Papers by Jitka Machalová
Analysis of Berger nonlinear elastic static plate bending of rectangular plates
Mathematics and Mechanics of Solids
This paper deals with a nonlinear static plate model based on Berger theory, which is a specific ... more This paper deals with a nonlinear static plate model based on Berger theory, which is a specific case of a generalization of the Woinowsky-Krieger mathematical model of beam bending. It is considered a plate bending with forces acting in the middle plane of the plate and a contact problem with an elastic foundation, where the normal compliance condition is employed. A variational equation of the problem and a functional of the total potential energy corresponding to the variational equation are derived. Under additional assumptions on the data (e.g., clamped plate), the existence and uniqueness of the solution are proved. A numerical solution is based on the Galerkin method and Courant approximation. The theory is illustrated by a numerical example.
arXiv (Cornell University), May 16, 2019
In the context of functional data analysis, probability density functions as nonnegative function... more In the context of functional data analysis, probability density functions as nonnegative functions are characterized by specific properties of scale invariance and relative scale which enable to represent them with the unit integral constraint without loss of information. On the other hand, all these properties are a challenge when the densities need to be approximated with spline functions, including construction of the respective spline basis. The Bayes space methodology of density functions enables to express them as real functions in the standard L 2 space using the centered log-ratio transformation. The resulting functions satisfy the zero integral constraint. This is a key to propose a new spline basis, holding the same property, and consequently to build a new class of spline functions, called compositional splines, which can approximate probability density functions in a consistent way. The paper provides also construction of smoothing compositional splines and possible orthonormalization of the spline basis which might be useful in some applications. Finally, statistical processing of densities using the new approximation tool is demonstrated in case of simplicial functional principal component analysis with anthropometric data.
Identification problem is a framework of mathematical problems dealing with the search for optima... more Identification problem is a framework of mathematical problems dealing with the search for optimal values of the unknown coefficients of the considered model. Using experimentally measured data, the aim of this work is to determine the coefficients of the given differential equation. This paper deals with the extension of the continuous dependence results for the Gao beam identification problem with different types of boundary conditions by using appropriate analytical inequalities with a special attention given to the Wirtinger's inequality and its modification. On the basis of these results for the different types of the boundary conditions the existence theorem for the identification problem can be proven.
PeerJ, Jan 14, 2019
Background. Seed germination is one of the earliest key events in the plant life cycle. The timin... more Background. Seed germination is one of the earliest key events in the plant life cycle. The timing of transition from seed to seedling is an important developmental stage determining the survival of individuals that influences the status of populations and species. Because of wide geographical distribution and occurrence in diverse habitats, wild pea (Pisum sativum subsp. elatius) offers an excellent model to study physical type of seed dormancy in an ecological context. This study addresses the gap in knowledge of association between the seed dormancy, seed properties and environmental factors, experimentally testing oscillating temperature as dormancy release clue. Methods. Seeds of 97 pea accessions were subjected to two germination treatments (oscillating temperatures of 25/15 • C and 35/15 • C) over 28 days. Germination pattern was described using B-spline coefficients that aggregate both final germination and germination speed. Relationships between germination pattern and environmental conditions at the site of origin (soil and bioclimatic variables extracted from World-Clim 2.0 and SoilGrids databases) were studied using principal component analysis, redundancy analysis and ecological niche modelling. Seeds were analyzed for the seed coat thickness, seed morphology, weight and content of proanthocyanidins (PA). Results. Seed total germination ranged from 0% to 100%. Cluster analysis of germination patterns of seeds under two temperature treatments differentiated the accessions into three groups: (1) non-dormant (28 accessions, mean germination of 92%), (2) dormant at both treatments (29 acc., 15%) and (3) responsive to increasing temperature range (41 acc., with germination change from 15 to 80%). Seed coat thickness differed between groups with dormant and responsive accessions having thicker testa (median 138 and 140 µm) than non-dormant ones (median 84 mm). The total PA content showed to be higher in the seed coat of dormant (mean 2.18 mg g −1) than those of non-dormant (mean 1.77 mg g −1) and responsive accessions (mean 1.87 mg g −1). Each soil and bioclimatic variable and also germination responsivity (representing
arXiv (Cornell University), May 16, 2019
In the context of functional data analysis, probability density functions as nonnegative function... more In the context of functional data analysis, probability density functions as nonnegative functions are characterized by specific properties of scale invariance and relative scale which enable to represent them with the unit integral constraint without loss of information. On the other hand, all these properties are a challenge when the densities need to be approximated with spline functions, including construction of the respective B-spline basis. The Bayes space methodology of density functions enables to express them as real functions in the standard L 2 space using the clr transformation. The resulting functions possess the zero integral constraint. This is a key to propose a new B-spline basis, holding the same property, and consequently to build a new class of spline functions, called simplicial splines, which can approximate probability density functions in a consistent way. The paper provides also construction of smoothing simplicial splines and possible orthonormalization of the B-spline basis which might be useful in some applications. Finally, a concise analysis is demonstrated with anthropometric data.
This contribution deals with a nonlinear beam model which was published by Gao in [1] and it is k... more This contribution deals with a nonlinear beam model which was published by Gao in [1] and it is known as the Gao beam. Let us denote E the Young's modulus, I the area moment of inertia, q the distributed transverse load, ν the Poissons ratio, L the length of the beam, 2h its height, while its width is b and it is considered as a unit. An axial force P is considered acting at the point x = L. The Gao beam model is described by the fourth-order differential equation EI w − Eα(w) 2 w + P µ w = f in (0, L),
Programs and Algorithms of Numerical Mathematics 21
Identification problem is a framework of mathematical problems dealing with the search for optima... more Identification problem is a framework of mathematical problems dealing with the search for optimal values of the unknown coefficients of the considered model. Using experimentally measured data, the aim of this work is to determine the coefficients of the given differential equation. This paper deals with the extension of the continuous dependence results for the Gao beam identification problem with different types of boundary conditions by using appropriate analytical inequalities with a special attention given to the Wirtinger's inequality and its modification. On the basis of these results for the different types of the boundary conditions the existence theorem for the identification problem can be proven.
In this contribution we deal with using of the optimal control method for solution of an inverse ... more In this contribution we deal with using of the optimal control method for solution of an inverse problem for nonlinear Gao beam which was published by Gao in [1]. The inverse problem will be formulated as the minimization of a cost functional which depends on a state problem. The state problem is represented by the Gao beam equation σIu − σα(u′)2u′′ = f in (0, L), u(0) = u′(0) = 0, u(L) = u′(L) = 0, (1)
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
arXiv: Classical Physics, 2019
In this small comment mathematical formulations concerning the nonlinear beam model published in ... more In this small comment mathematical formulations concerning the nonlinear beam model published in [1] are analyzed. The beam is subjected to vertical and axial loading (at its right end). This nonlinear model can be used to study post-buckling problems. Unfortunately, some inconsistency between pure bending and pure buckling problems was discovered by the authors of this comment. This is concerned with definition of an integral constant, which is not in [1] strictly determined. In this comment there is proposed the adjustment of of this constant which should solve the mentioned troubles.
In this paper are given definition of Chipman pseudoinverse, which is generalized Moore-Penrose m... more In this paper are given definition of Chipman pseudoinverse, which is generalized Moore-Penrose matrix, some its properties and algorithms for its computation. One example for its application in spline theory is shown at the end.
A new orthogonal decomposition for bivariate probability densities embedded in Bayes Hilbert spac... more A new orthogonal decomposition for bivariate probability densities embedded in Bayes Hilbert spaces is derived. It allows one to represent a density into independent and interactive parts, the former being built as the product of revised definitions of marginal densities and the latter capturing the dependence between the two random variables being studied. The developed framework opens new perspectives for dependence modelling (which is commonly performed through copulas), and allows for the analysis of dataset of bivariate densities, in a Functional Data Analysis perspective. A spline representation for bivariate densities is also proposed, providing a computational cornerstone for the developed theory.
where |x1| = |x2| = |x3| = m, n = 3m, A ∈ Rn×n is the symmetric, positive definite Hessian matrix... more where |x1| = |x2| = |x3| = m, n = 3m, A ∈ Rn×n is the symmetric, positive definite Hessian matrix, b ∈ R, and l, g ∈ R. This problem arises, e.g., in duality based methods for the solution of 3D contact problems of linear elasticity with Tresca friction. As a widely used approach of contact problems with (more realistic) Coulomb friction is based on a sequence of Tresca friction problems [2], an efficient solver for (1) is of crucial importance. In this contribution we shall test algorithms based on an ”interior point” idea.
Mathematics, 2020
This paper deals with the identification of coefficients in the nonlinear beam model which was fi... more This paper deals with the identification of coefficients in the nonlinear beam model which was first introduced by D. Y. Gao in 1996. For the identification of coefficients, an optimal control approach is used. The unknown coefficients are material parameters of the beam and play the role of the control variables. The existence of at least one solution of the optimal control problem is proved. For the studied problem the finite element approximation is provided. Finally some illustrative examples are introduced.
Optimal thickness distribution of stepped nonlinear Gao beam
Mathematics and Computers in Simulation, 2020
Abstract We present an optimal shape design problem of a transversely and axially loaded nonlinea... more Abstract We present an optimal shape design problem of a transversely and axially loaded nonlinear stepped elastic beam with piecewise constant thickness distribution. The nonlinear beam model proposed by D.Y. Gao in 1996 is considered for modelling the state problem. The objective of the optimization is to find the thickness distribution to maximize the stiffness of the beam subjected to geometrical constraints, namely the preservation of volume is imposed. The stiffness is characterized by the compliance cost functional. The existence of solutions of the optimization problem and convergence properties of discretization using finite element approach is analysed. The theory is illustrated by numerical results.
Mathematics and Mechanics of Solids, 2017
This paper presents mathematical formulations and a solution for contact problems that concern th... more This paper presents mathematical formulations and a solution for contact problems that concern the nonlinear beam published by Gao (Nonlinear elastic beam theory with application in contact problems and variational approaches, Mech Res Commun 1996; 23: 11–17) and an elastic foundation. The beam is subjected to a vertical and also axial loading. The elastic deformable foundation is considered at a distance under the beam. The contact is modeled as static, frictionless and using the normal compliance contact condition. In comparison with the usual contact problem formulations, which are based on variational inequalities, we are able to derive for our problem a nonlinear variational equation. Solution of this problem is realized by means of the so-called control variational method. The main idea of this method is to transform the given contact problem to an optimal control problem, which can provide the requested solution. Finally, some results including numerical examples are offered ...
Comments on the large deformation elastic beam model developed by D.Y. Gao
Mechanics Research Communications, 2020
Abstract In this comment mathematical formulations concerning the nonlinear beam model published ... more Abstract In this comment mathematical formulations concerning the nonlinear beam model published in [1] are analysed. The beam is subjected to vertical and axial loading (at its right end). This model can be especially used to study post-buckling problems, but in this paper we focus only on convex cases. However, some inconsistency between properties of bending problems and their convexity bounds was discovered by the authors of this comment. This is concerned with definition of an integral constant, which is in [1] not strictly determined, only later in [3]. In this comment there is proposed an alternative definition of this constant which should solve the mentioned discrepancy. Except this one, no in-depth revision of the considered beam model is intended here.
A comparison of seed germination coefficients using functional regression
Applications in Plant Sciences, 2020
Premise Seed germination over time is characterized by a sigmoid curve, called a germination curv... more Premise Seed germination over time is characterized by a sigmoid curve, called a germination curve, in which the percentage (or absolute number) of seeds that have completed germination is plotted against time. A number of individual coefficients have been developed to characterize this germination curve. However, as germination is considered to be a qualitative developmental response of an individual seed that occurs at one time point, but individual seeds within a given treatment respond at different time points, it has proven difficult to develop a single index that satisfactorily incorporates both percentage and rate. The aim of this paper is to develop a new coefficient, the continuous germination index (CGI), which quantifies seed germination as a continuous process, and to compare the CGI with other commonly used indexes. Methods To create the new index, the germination curves were smoothed using nondecreasing splines and the CGI was derived as the area under the resulting spline. For the comparison of the CGI with other common indexes, a regression model with functional response was developed. Results Using both an experimentally obtained wild pea (Pisum sativum subsp. elatius) seed data set and a hypothetical data set, we showed that the CGI is able to characterize the germination process better than most other indices. The CGI captures the local behavior of the germination curves particularly well. Discussion The CGI can be used advantageously for the characterization of the germination process. Moreover, B‐spline coefficients extracted by its construction can be employed for the further statistical processing of germination curves using functional data analysis methods.
Plants, 2020
Seed dormancy and timing of its release is an important developmental transition determining the ... more Seed dormancy and timing of its release is an important developmental transition determining the survival of individuals, populations, and species in variable environments. Medicago truncatula was used as a model to study physical seed dormancy at the ecological and genetics level. The effect of alternating temperatures, as one of the causes releasing physical seed dormancy, was tested in 178 M. truncatula accessions over three years. Several coefficients of dormancy release were related to environmental variables. Dormancy varied greatly (4–100%) across accessions as well as year of experiment. We observed overall higher physical dormancy release under more alternating temperatures (35/15 °C) in comparison with less alternating ones (25/15 °C). Accessions from more arid climates released dormancy under higher experimental temperature alternations more than accessions originating from less arid environments. The plasticity of physical dormancy can probably distribute the germination...
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Papers by Jitka Machalová